Problem 182 - Calculus Volume 1

Question

The function \(H(t)=8 \sin \left(\frac{\pi}{6} t\right)\) models the height \(H(\text { in feet })\) of the tide \(t\) hours after midnight. Assume that \(t=0\) is midnight. a. Find the amplitude and period. b. Graph the function over one period. c. What is the height of the tide at 4:30 a.m.?

Step-by-Step Solution

Answer

Amplitude: 8, Period: 12 hours, Height at 4:30 a.m.: approximately 5.66 feet.

Step 1: Identify the Amplitude

The amplitude of a sinusoidal function of the form \(y = a \sin(bt)\) is the absolute value of the coefficient \(a\). In this case, the given function is \(H(t) = 8 \sin \left(\frac{\pi}{6} t\right)\). Hence, the amplitude is \(8\).

Step 2: Calculate the Period

The period of a sinusoidal function \(y = a \sin(bt)\) is given by \(\frac{2\pi}{b}\). In this function, \(b\) is \(\frac{\pi}{6}\). Thus, the period is calculated as: \[ \text{Period} = \frac{2\pi}{\frac{\pi}{6}} = 12 \text{ hours}.\]

Step 3: Graph the Function

To graph the function over one period, plot the function \(H(t) = 8 \sin \left(\frac{\pi}{6} t\right)\) from \(t=0\) to \(t=12\) hours. The graph will oscillate between \(-8\) and \(8\) with one full cycle, starting at \(H(0) = 0\), reaching the maximum \(H(t) = 8\) at \(t=3\), crossing zero at \(t=6\), reaching the minimum \(H(t) = -8\) at \(t=9\), and ending at zero again at \(t=12\).

Step 4: Determine Height at 4:30 a.m.

Calculate the height by substituting \(t = 4.5\) (since 4:30 a.m. is 4.5 hours after midnight) into the function: \[ H(4.5) = 8 \sin \left(\frac{\pi}{6} \times 4.5\right) = 8 \sin \left(\frac{4.5\pi}{6}\right) = 8 \sin \left(\frac{3\pi}{4}\right). \] Use the sine value \(\sin\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2}\). Therefore, \[ H(4.5) = 8 \times \frac{\sqrt{2}}{2} = 4\sqrt{2}. \] Approximately, \(H(4.5)\) is about \(5.66\) feet.

Key Concepts

Amplitude

The amplitude of a trigonometric function like the sine function is a measure of its peak value. In simple terms, it shows how high or low the wave reaches from its central position. Amplitude is crucial for understanding how "tall" the wave is.

For any sinusoidal function of the form \(y = a \sin(bt)\), the amplitude is given by the absolute value of the coefficient \(a\). Here, it refers to the maximum deviation from the middle of the wave.

• The given function is \(H(t) = 8 \sin \left(\frac{\pi}{6} t\right)\).
• The amplitude is \(|8|\), which is simply \(8\).

Thus, in the context of this tidal function, the tide can rise or fall by 8 feet from the mean sea level. This means that the wave reaches its peak at 8 feet above and its lowest at 8 feet below the central point, indicating the total vertical range is 16 feet.

This measure helps us understand the intensity of the tidal changes during a single cycle.

Period

The period of a trigonometric function describes how long it takes for the function to complete one full cycle. It's like telling us the "length" of one wave cycle in terms of time or horizontal distance. A shorter period means more waves in less time, and a longer period signifies fewer cycles over the same duration.

For the sinusoidal function \(y = a \sin(bt)\), the period \(T\) is calculated using the formula \(\frac{2\pi}{b}\). Given the function \( H(t) = 8 \sin \left(\frac{\pi}{6} t\right) \), we identify:

• \(b = \frac{\pi}{6}\).
• The period \(T = \frac{2\pi}{\frac{\pi}{6}} = 12\) hours.

This tells us that it takes 12 hours for the tidal wave to repeat its pattern, starting at a particular height at midnight and returning to the same height at noon.

In practical terms, the tide will experience a full high and low cycle within 12 hours, reflecting how often certain tidal activities might occur during a 24-hour period.

Sinusoidal Functions

Sinusoidal functions, such as sine and cosine, form the backbone of trigonometric graphs. They are periodic and repeat their cycle over consistent intervals, making them perfect for modeling cyclical phenomena like sound waves, electrical currents, and in this context, tides.

The typical structure of a sinusoidal function is \(y = a \sin(bt)\), where:

• \(a\) represents the amplitude, dictating the wave's height.
• \(b\) influences the period, determining the cycle length.

The function \(H(t) = 8 \sin \left(\frac{\pi}{6} t\right)\) specifically models the rise and fall of tides.

• A maximum height is reached when the sine function peaks at 1, pushing the height to 8 feet above the mean.
• A minimum height occurs when the sine function hits -1, lowering the tide to 8 feet below its central point.
• At points where the sine function crosses zero, the tide aligns with the mean sea level.

Understanding sinusoidal functions helps in predicting these regular patterns, crucial for navigation, fishing, and planning by coastal communities.

They tell us not just how the tides move, but also predict when and by how much the water levels will change over the day.